Conic Bundle

In algebraic geometry, a conic bundle is an algebraic variety that appears as a solution of a Cartesian equation of the form

: $X^2 + aXY + b Y^2 = P (T).\,$

Theoretically, it can be considered as a Severi–Brauer surface, or more precisely as a Châtelet surface. This can be a double covering of a ruled surface. Through an isomorphism, it can be associated with a symbol $(a, P)$ in the second Galois cohomology of the field $k$.

In fact, it is a surface with a well-understood divisor class group and simplest cases share with Del Pezzo surfaces the property of being a rational surface. But many problems of contemporary mathematics remain open, notably (for those examples which are not rational) the question of unirationality.

A naive point of view

To write correctly a conic bundle, one must first reduce the quadratic form of the left hand side. Thus, after a harmless change, it has a simple expression like

: $X^2 - aY^2 = P (T). \,$

In a second step, it should be placed in a projective space in order to complete the surface "at infinity".

To do this, we write the equation in homogeneous coordinates and expresses the first visible part of the fiber

: $X^2 - aY^2 = P (T) Z^2. \,$

That is not enough to complete the fiber as non-singular (smooth and proper), and then glue it to infinity by a change of classical maps:

Seen from infinity, (i.e. through the change $T\mapsto T'=\frac 1 T$), the same fiber (excepted the fibers $T = 0$ and $T '= 0$), written as the set of solutions $X'^2 - aY'^2= P^* (T') Z'^2$ where $P^* (T ')$ appears naturally as the reciprocal polynomial of $P$. Details are below about the map-change ':y': z '/math>.

The fiber ''c''

Going a little further, while simplifying the issue, limit to cases where the field $k$ is of characteristic zero and denote by $m$ any integer except zero. Denote by ''P''(''T'') a polynomial with coefficients in the field $k$, of degree 2''m'' or 2''m'' − 1, without multiple root. Consider the scalar ''a''.

One defines the reciprocal polynomial by $P^*(T')=T^P(\frac 1 T)$, and the conic bundle ''F''_''a'',''P'' as follows :

Definition

$F_$ is the surface obtained as "gluing" of the two surfaces $U$ and $U'$ of equations

:$X^2 - aY^ 2 = P (T) Z^2$

and

:$X '^2 - aY'^2 = P (T ') Z'^ 2$

along the open sets by isomorphisms

:$x '= x,, y' = y,$ and $z '= z t^m$.

One shows the following result :

Fundamental property

The surface ''F''_''a'',''P'' is a ''k'' smooth and proper surface, the mapping defined by

:$p: U \to P_$

by

:( :y:zt)\mapsto t

and the same on $U '$ gives to ''F''_''a'',''P'' a structure of conic bundle over ''P''_1,''k''.

See also

* Algebraic surface
* Intersection number (algebraic geometry)
* List of complex and algebraic surfaces

References

*
*
*{{cite book
, author = David Eisenbud
, author-link = David Eisenbud
, year = 1999
, title = Commutative Algebra with a View Toward Algebraic Geometry
, publisher = Springer-Verlag
, isbn = 0-387-94269-6

Algebraic geometry
Algebraic varieties